Prove the theorem: Given: line p is tangent to circle with center O, H is the point of tangency. Prove: p ⊥ OH.

Proof of the Theorem:

We need to prove that the tangent line p is perpendicular to the radius OH at the point of tangency H.

Given: Line p is tangent to circle with center O, H is the point of tangency.

To Prove: p ⊥ OH.

Proof:

1. Let A be any point on the tangent line p, other than H.
2. Since p is tangent to the circle at H, every other point A on p must lie outside the circle.
3. Therefore, the distance OA must be greater than the radius R (which is equal to OH). So, OA > OH.
4. This means that OH is the shortest distance from the center O to the line p.
5. By definition, the shortest distance from a point to a line is the perpendicular distance.
6. Hence, the line segment OH must be perpendicular to the tangent line p.
7. Therefore, p ⊥ OH.

Theorem proved.